The Miracles of Mathematics Will Make Your World More Vibrant

Photo, Twenty20.
“Why does mathematics describe nature? That’s a deeper question than most.” –Terence McKenna

When I was living in Egypt for a year, I took a series of math courses in pursuit of a PhD. I was never “talented” at math in K-12, but research shows that the way math is taught inside public education tends to destroy students’ capacity to learn or wield math later in life. But that’s another story…

Thankfully, before I pulled the trigger on the PhD, I talked to recent graduates of the program. I found they were all five or six years older than me, broke, cynical, and none of them were excited that they had “invested” their time in their PhD — whew! Bullet dodged.

Through the self-directed struggle of seeking to understand mathematics, I inevitably turned to Wolfram Alpha for help.

For those who think life is boring, humdrum, or dull… I’d like to present a small solution. Enter one (or all) of the five gateways below, and explore mathematical concepts on your own. You don’t have to “understand” or grok them right away. As you just hold them in your mind, and revisit them over time.

You will find that your world becomes more mysterious and your life becomes a chaotic adventure. You might even find yourself emerged in a mystery.

Wolfram Alpha and its Inventor

Instead of a calculator, try using the conversational Wolfram Alpha. Wolfram Alpha is an amazing computational engine that (among many other things) helps take problems presented in our natural language and convert them into mathematical problems and solutions.

This led me to the shores of the mathematical iceberg. I stared into the waters below and could only glimpse at how massive the portion of it was that was submerged.

The founder of WA, Stephen Wolfram, is a fascinating person, and you can get one of his books for free here.

“It is perhaps a little humbling to discover that we as humans are in effect computationally no more capable than cellular automata with very simple rules. But the Principle of Computational Equivalence also implies that the same is ultimately true of our whole universe.
So while science has often made it seem that we as humans are somehow insignificant compared to the universe, the Principle of Computational Equivalence now shows that in a certain sense we are at the same level as it is. For the principle implies that what goes on inside us can ultimately achieve just the same level of computational sophistication as our whole universe.” ― Stephen Wolfram

The Golden Ratio in Nature and Music

Photo, Twenty20.
“The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet.” — LiveScience

You see the beauty of the pinecone. The height of the second layer compared to the height of the first is equivalent to the summed heights of the first and second layers divided by the height of the second layer (the golden ratio).

Here is what the golden ratio sounds like in music.

Now when you look out into the world or hear a melody, you might spot an underlying structure of order. The “divine proportion” is even present in human faces, eyes, and teeth.

The Fibonacci Sequence

Photo, Twenty20.
“In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones:” — Wikipedia

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

You’re probably thinking, “Great. But why do these numbers matter?” Well, these numbers have many real-world applications, including finance. Many traders and hedge funds use mathematical models that incorporate the fibonacci sequence in an attempt to generate returns.

Check out this TED Talk for a gateway into the Fibonacci sequence

Chaos Theory and Ralph Abraham

“Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. ‘Chaos’ is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns… The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a tornado in Texas.” — Wikipedia

Basically, Chaos Theory says that you’ll never be able to predict the exact future.

Chaos theory… says that you can never predict the weather more than a few days away. All the money that has been spent on long-range forecasting — about half a billion dollars in the last few decades — is money wasted. It’s a fool’s errand. It’s as pointless as trying to turn lead into gold. We look back at the alchemists and laugh at what they were trying to do, but future generations will laugh at us the same way. –Michael Crichton

Ralph Abraham is a professor, mathematician, and chaos theorist whose work has focused on the intersection of mathematics and psychedelics since the 1960s.

My specific goal is to revolutionize the future of the species. Mathematics is just another way of predicting the future. –Ralph Abraham

His paper called Mathematics and Mysticism is an engaging dive into what math might mean. From the abstract:

Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.

How Fractals and the Genius of Mandelbrot Can Relieve Your Anxiety

Photo by By Binette228 — Own work, CC BY-SA 3.0

A fractal is a curve or geometric figure, with each part having the same statistical character as the whole.

These fractal sets were named after Benoit Mandelbrot, whose pioneering work in developing what is now known as “fractal geometry” paved the way for many more mathematicians to study and expound on his ideas.

“Because of his access to IBM’s computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. He showed how visual complexity can be created from simple rules. He said that things typically considered to be “rough”, a “mess” or “chaotic”, like clouds or shorelines, actually had a ‘degree of order.’”

The visual representations of Mandelbrot sets are stunning.

The next time you think your life is too chaotic or a mess, just keep persisting. Mathematics has proven that there is an implicate order in everything around us. If the golden ratio is present in the layout of our faces and in everything around us… then there is also an underlying (albeit unseen) order and structure present in our lives. It’s okay to drop certain anxieties and trust what mathematics shows us- there is an underlying order.


The gateways into the miracles of mathematics are all around us.

The universe is not only stranger than we imagine, it is stranger than we can imagine. — JBS Haldane

Whether you use these gateways to bring more meaning to your life, balance your budget, or revel in the mystery… they’ll open up new possibilities and options you didn’t know existed.


Chad Grills is the founder of The Mission, your #1 source for accelerated learning. You can subscribe to their M-F newsletter here.

If you enjoyed this article, please recommend it and share to help others find it. Leave a comment below and let me know who or what you love about math!